Pak K. Yuet M.Sc., Chemical Engineering (1990) Queen's University B.Eng., Chemical Engineering (1988) Technical University of Nova Scotia

Transcription

1 61/ Theoretical and Experimental Studies of Vesicle Formation in Surfactant Mixtures by Pak K. Yuet M.Sc., Chemical Engineering (1990) Queen's University B.Eng., Chemical Engineering (1988) Technical University of Nova Scotia Submitted to the Department of Chemical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1996 Massachusetts Institute of Technology All rights reserved. Author. Certified by Department of Chemical Engineering May 16, 1996 Daniel Blankschtein Associate Professor Thesis Supervisor Accepted by Robert E. Cohen Chairman, Departmental Committee on Graduate Students OF TECHNOLCGY JUN Science

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3 Theoretical and Experimental Studies of Vesicle Formation in Surfactant Mixtures by Pak K. Yuet M.Sc., Chemical Engineering (1990) Queen's University B.Eng., Chemical Engineering (1988) Technical University of Nova Scotia Submitted to the Department of Chemical Engineering on May 16, 1996, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract A fundamental understanding of vesicle formation and stability in mixed surfactant systems is important for the description of their phase behavior, for the application of vesicles as 'encapsulating devices, and for the elucidation of cholesterol gallstone formation in bile, where the solubilization of cholesterol in vesicles has been suggested to correlate with the metastability of bile. To gain a better understanding of vesiculation, a molecular-thermodynamic theory was developed to describe the formation of mixed surfactant vesicles. The theory incorporates a detailed modeling of the various free-energy contributions associated with vesiculation, including a mean-field calculation of the free-energy contribution associated with packing of the surfactant tails in the vesicle bilayer, an estimation of the surfactant-head steric repulsions using an equation of state for hard-disk mixtures, in the context of the scaled-particle theory, and a calculation of the electrostatic free energy of a vesicle using approximate analytical expressions for the surface potentials. By knowing only the molecular structure of the surfactants involved in vesicle formation and the solution conditions, the theory can predict a wealth of vesicle properties, including vesicle size and composition distribution, surface charge densities, surface potentials, and compositions of vesicle leaflets. More importantly, this theory is able to reveal the relative importance of, as well as the interplay between, the various free-energy contributions to vesiculation. In particular, it was found that: (i) the distribution of surfactant molecules between the two vesicle leaflets plays a major role in vesiculation, (ii) vesicles that are stabilized by the entropy of mixing, such as those containing surfactants of similar tail lengths,

4 are large and widely distributed in size, and (iii) mixed surfactant vesicles, which are characterized by small sizes and a narrow size distribution, can be stabilized energetically in highly asymmetric surfactant mixtures. In addition, it was found that vesicle composition is determined by the interplay between the entropy of mixing and the free energy of vesiculation. Specifically, decreasing surfactant tail-length asymmetry reduces the energetic influence, as compared to the entropic one, by decreasing the surfactant tail transfer free energy, thus producing an effect on vesicle composition similar to that produced by adding salt, which reduces the electrostatic free-energy contribution associated with vesiculation. In the experimental study of cholesterol solubilization in model bile, a systematic comparison of ultracentrifugation and gel chromatography was first conducted regarding their ability to separate vesicles and mixed micelles in model bile. It was found that ultracentrifugation overestimates vesicular cholesterol content, mainly due to the creation of a bile salt depletion zone. This technique was then modified by reducing the mobility of mixed micelles in a centrifugal field. The distribution of cholesterol measured by the modified technique was found to agree well with that measured using gel chromatography. This modified technique and gel chromatography were then used in a two-level factorial experiment to investigate the effects of several physiological variables, including total lipid content, bile salt (BS) to egg-yolk phosphatidylcholine (EYPC) ratio [BS/(BS+EYPC)], cholesterol (Ch) content, and bile salt hydrophobicity, on two responses: the distribution of cholesterol and the vesicular Ch/EYPC ratio. The results show that: (i) the total lipid content has a significant but opposite effect on the two responses, (ii) increasing total lipid content reduces the percentage of cholesterol in vesicles while raising the vesicular Ch/EYPC ratio, (iii) the BS/(BS+EYPC) ratio is the most important variable in determining the vesicular Ch/EYPC ratio, but does not seem to affect the distribution of cholesterol, and (iv) the bile salt hydrophobicity affects both responses, presumably through the interactions with the hydrophobic moieties of the phospholipids. Total lipid content was also found to interact strongly with the BS/(BS+EYPC) ratio and with the bile salt hydrophobicity in determining the distribution of cholesterol. In addition, the effect of bile salt hydrophobicity on the vesicular Ch/EYPC ratio was found to depend on total lipid content, as well as on the BS/(BS+EYPC) ratio. These findings suggest that the metastability of bile may be correlated to the vesicular Ch/EYPC ratio, but not to the distribution of cholesterol, and that it is possible to effectively alter the two responses by varying several physiological variables simultaneously. The theoretical and experimental findings of this thesis should contribute to our fundamental knowledge on surfactant mixtures, as well as on the formation of cholesterol gallstones in bile. It is also hoped that this thesis will serve as a gateway for further exciting and rewarding discoveries in the area of mixed surfactant systems. Thesis Supervisor: Daniel Blankschtein Title: Associate Professor

5 Acknowledgments I dedicate this thesis to my parents. Without their encouragement, I would not have pursued my Ph.D. degree. I thank them for their love; I thank them for everything. I would also like to thank Professor Blankschtein for his guidance and advice. He is always there when I need him, both academically and personally. In fact, after six years, he is more than just a thesis advisor; he is a good friend of mine. And, of course, Ginger is always there to listen to my complaints and to talk to me; without her, my final year at MIT would have been very different. Thanks also go to Dr. Joanne Donovan for her advice on the experimental program of this thesis, to Dr. Sudhakar Puvvada for his help in getting me started on this work, to Irene Kotok for her assistance in the experimental work, to Monika Leonard for teaching me the laboratory skills, and to Dr. Martin Carey for many helpful discussions. And now the crew. I would like to thank Dr. Leo Lue for the many late-night intellectual discussions, the Ashdown desserts, and some very "deep" conversations on the meaning of life, Nancy Zoeller for sharing the cubicle for over three years (without complaint), Anat Shiloach for the coffee run, Dr. Chia-li Liu for just being there (when I thought nobody was around), and Ayal Naor for introducing me to the "real" music. Finally, I would like to express my gratitude to the National Institute of Health (NIH) Biotechnology Training Grant and the American Liver Foundation for financial support.

6 Contents 1 Introduction 1.1 Self-Assembly of Surfactants Biomedical Implication of Vesicles Bile and Cholesterol Gallstones Model Biliary System Research Motivation Theoretical Studies of Mixed Vesicles Experimental Studies of Biliary System Research Objectives... 2 Molecular-Thermodynamic Theory of Mixed Vesicles 2.1 Thermodynamic Framework to Describe a Vesicle Suspension. 2.2 Molecular Model of Vesicle Formation Transfer Free Energy Packing Free Energy Interfacial Free Energy Steric Free Energy Electrostatic Free Energy Computational Procedure Concluding Remark

7 3 Approximate Expressions for the Surface Potentials of Charged Vesicles 3.1 Implicit Relations between Surface Potentials and Surface Charge Densities Approximate Analytical Expressions for the Surface Potentials Results and Discussions Concluding Remarks... 4 Application of the Theory: I. Cationic/Anionic Surfactant Mixture 4.1 M odel System Free Energy of Vesiculation Size and Composition Distribution Effect of Added Salt Concluding Remarks Application of the Theory: II. Effect of Surfactant Tail-Length Asymmetry on the Formation of Mixed Surfactant Vesicles Model Systems and Molecular Parameters Effect of Surfactant Tail-Length Asymmetry on Vesicle Composition Effect of Surfactant Tail-Length Asymmetry on Vesicle Size Concluding Remarks Separation of Biliary Aggregates Materials and Methods Model Bile Preparation Ultracentrifugation Gel Chromatography Lipid Analysis Quasi-Elastic Light Scattering R esults Centrifugal Separation of Mixed Micelles and Vesicles

8 6.2.2 Comparison between Ultracentrifugation and Gel Chromatography Phase Alteration During Ultracentrifugation Discussion Concluding Remarks Factorial Experimental Study of Cholesterol Distribution and Vesicle Composition 7.1 Materials and Methods Model Bile Preparation Modified Ultracentrifugation Separation of Vesicles and Mixed Micelles 7.2 Statistical Experimental Design Response and Process Variables Two-Level Factorial Design Results Modified Ultracentrifugation Distribution of Cholesterol Vesicular Ch/EYPC Ratio D iscussion Concluding Remarks Conclusions and Future Research Directions 8.1 Thesis Sum m ary Future Directions for Theoretical Work Molecular Model of Vesiculation Entropy of Mixing, Gm, and Interaction Free Energy, Gint Global Phase Behavior of Surfactant Mixtures Application to the Biliary System Future Directions for Experimental Work Non-Linear Behavior - Higher-Order Design

9 8.3.2 Thermodynamic Activity of Cholesterol Concluding Remarks A Size and Composition Distribution B Chain Packing in a Vesicle C Steric Free Energy D Electrostatic Free Energy E Summary of Model Equations F Geometric Constraints in a Vesicle G Derivation of Analytical Expressions for the Surface Potentials

10 List of Figures 1-1 Schematic diagram of a two-component unilamellar surfactant vesicle Molecular structure of taurine-conjugated bile acids Molecular structure of phosphatidylcholine Molecular structure of cholesterol Schematic equilibrium phase diagram for the pseudo-ternary system containing taurocholate, egg-yolk phosphatidylcholine, cholesterol, and a fixed proportion of water Schematic representation of a two-component unilamellar vesicle Schematic diagram depicting the approximation used in the calculation of the electrostatic free energy Schematic diagram of a positively-charged vesicle Predicted variation of the free energy of vesiculation, g,,,, as a function of vesicle aggregation number, n, and composition, F Predicted variations of the transfer free energy, gtr, and the electrostatic free energy, gl,ec, as a function of F for a planar bilayer Predicted variation of the distribution of molecules, f, and the freeenergy difference, gves gbilayer, as a function of the dimensionless mean curvature, Predicted variations of the outer, ao, and inner, ai, areas per molecule as a function of the dimensionless mean curvature, i

11 4-5 Predicted variation of the free-energy contributions, gg, gsteric, Ypack, and gelec, as a function of the dimensionless mean curvature, Predicted size and composition distribution, X(n, F), for a CTAB/SOS aqueous system containing 2 wt% surfactant and a CTAB/SOS ratio of 3/7 by weight Predicted effect of concentration of added salt on the outer surface potential and on the outer surface charge density of vesicles in the CTAB/SOS aqueous system Predicted effect of concentration of added salt on vesicle radius and on peak composition in vesicles, F* Predicted variation of g,,,es - g as a function of vesicle composition, F, for a large vesicle (n = 107) Predicted size and composition distribution, X(n, F), for a C16/C15 aqueous system containing 2 wt% surfactant and a C16/C15 ratio of 3/7 by weight Predicted size and composition distribution, X(n, F), for a C16/C5 aqueous system containing 2 wt% surfactant and a C16/C5 ratio of 3/7 by weight Predicted variation of the free-energy difference, gves -gbilayer, as a function of the dimensionless mean curvature, ý, for various cationic/anionic surfactant m ixtures Predicted methylene segment density distributions, (O(x')), for a C16 tail in the vesicle hydrophobic region of a C16/C15 vesicle having a = 0.37, F = 0.5, and f = Predicted lateral pressure in the hydrophobic region of a C16/C15 vesicle having ý = 0.37, F = 0.5, and f = 0.7, and a C16/C5 vesicle having E = 0.43, F = 0.5, and f =

12 5-7 Predicted variation of the free-energy contributions, ga, gsteric, and gpack, for a C16/C15 mixture as a function of the dimensionless mean curvature, Predicted segment density distributions, (O(x')), for a C16 tail in the vesicle hydrophobic region of a C16/C5 vesicle having 6 = 0.43, F = 0.5, and f = Predicted order parameters, Sz, for a C16 tail in a C16/C15 vesicle having 6 = 0.37, F = 0.5, and f = 0.7, and a C16/C5 vesicle having c = 0.43, F = 0.5, and f = Predicted variation of the free-energy contributions, ga, gsteric, and gpack, for a C16/C5 mixture as a function of dimensionless mean curvature, Effect of varying the density of the medium on the distribution of EYPC in micellar biles after 8 hrs of ultracentrifugation Distributions of Ch and EYPC in a vesicle suspension and micellar bile after ultracentrifugation for 13 hrs at a density of 1.03 g/ml Distribution of bile salt among the four fractions in a simple micellar solution after ultracentrifugation Effect of duration of ultracentrifugation and incubation on the percent Ch in vesicles Cholesterol elution profile of model bile before and after ultracentrifugation B-1 Predicted variation of the packing free energy of a planar bilayer, gpoack, containing C16 and C8 tails as a function of vesicle composition, F, and bilayer thickness, tb D-1 Comparison between the predicted electrostatic free energy per molecule obtained by using Eq. (D.1), g (4), and that obtained by using Eq. (2.27), g(2) 187 gelec....

13 . List of Tables 3.1 Comparison between the approximate solutions and the numerical integration of the PB equations for no = 0.1 M Comparison between the approximate solutions and the numerical integration of the PB equations for no = 0.01 M Comparison between the approximate solutions and the numerical integration of the PB equations for no = M Electrostatic free energies per molecule, gelec Molecular properties of cetyltrimethylammonium bromide (CTAB) and sodium octyl sulfate (SOS) Predicted values of some average vesicle properties in the CTAB/SOS aqueous system (2 wt% surfactant, CTAB/SOS = 3/7 by weight) Molecular properties of cetyltrimethylammonium bromide (CTAB), sodium pentadecyl sulfate (SPDS), sodium octyl sulfate (SOS), and sodium pentyl sulfate (SPS) EYPC concentrations in the four fractions after ultracentrifugation at various medium densities Percent of total Ch and Ch/EYPC ratio in vesicles as measured by ultracentrifugation and gel chromatography Percent of total Ch and Ch/EYPC ratio in vesicles as measured by modified ultracentrifugation and gel chromatography

14 7.2 Experimental conditions and measured responses for the 2 4 two-level factorial design High and low levels for the process variables Ch, EYPC, and TC distributions in vesicle suspension, micellar bile, and simple micellar solution, respectively, after centrifugation Estimated values of the coefficients for the distribution of cholesterol, Rch, and the vesicular Ch/EYPC ratio, obtained from the 2 4 design Average values of the distribution of cholesterol, Rch (%), at various experimental conditions Average values of the vesicular Ch/EYPC ratio at various experimental conditions B. 1 Packing free energy of a planar bilayer, pack (kt/molecule), containing C16 and C8 tails as a function of the vesicle composition, F, and the bilayer thickness, tb

15 Chapter 1 Introduction Surfactants, or surface active agents, are molecules that contain both hydrophilic ("water-loving") and hydrophobic ("water-fearing") moieties. The hydrophilic moiety, usually referred to as the "head", prefers to be surrounded by water molecules, whereas the hydrophobic moiety, usually referred to as the "tail", tends to repel water molecules. Because of this dual affinity (sometimes referred to as "amphiphilicity"), when surfactants are placed in water which is in contact with air, they migrate to the water/air surface, with their hydrophobic tails protruding into the air and their hydrophilic heads immersed in the water. Similarly, when surfactants are placed in a system containing water and hydrocarbon, they collect at the macroscopic water/hydrocarbon interface, with their hydrophobic tails now residing in the hydrocarbon phase. By collecting at surfaces or interfaces, surfactants have the ability to lower surface or interfacial tensions. This property has been widely exploited in detergents, shampoos, and other cleansing agents, which are an indispensable part of our modern life. The physical origin of the migration of surfactants to the surface of an aqueous system, or to the interface of a water/hydrocarbon system, is simply the minimization of the system free energy. When a surfactant molecule is placed in water, the water molecules surrounding the hydrophobic tail are forced to adopt a more ordered arrangement, as compared to that in pure water. By transferring the surfactant molecule to the surface or interface, and removing the tail out of the aqueous phase, the previously ordered water molecules can be released, thus gaining

16 entropy and lowering the free energy of the system. This is, indeed, the so-called "hydrophobic effect" [162]. 1.1 Self-Assembly of Surfactants As the surface or interface gets more crowded, it becomes increasingly difficult to transfer additional surfactant molecules to that location, and therefore, the freeenergy gain associated with this transfer process diminishes as the solution becomes more concentrated in surfactant. Beyond a certain threshold surfactant concentration, known as the critical micelle concentration (CMC) 1, therefore, the surfactant molecules prefer to self-assemble in the aqueous phase, forming microstructures that contain both hydrophobic and hydrophilic regions. The hydrophobic region is composed of the surfactant tails, and is shielded from water by the hydrophilic region composed of the surfactant heads. Self-assembling thus constitutes another vehicle to accommodate for the hydrophobic effect. Surfactants can self-assemble in dilute aqueous solutions into a variety of microstructures, including micelles, vesicles, and lamellae. In particular, unilamellar vesicles, which are composed of a closed bilayer that separates an inner aqueous compartment from the outer aqueous environment, are often found in various aqueous surfactant systems. Figure 1-1 shows a schematic representation of a two-component unilamellar surfactant vesicle, with the two types of surfactant heads represented by the black and white circles. Note that the vesicle hydrophobic region may be viewed as composed of an outer and an inner leaflet. The outer and inner leaflets correspond to those regions formed by the surfactant molecules anchoring at the outer and inner hydrocarbon/water interfaces, respectively. Because of their unique morphology, vesicles have been used as encapsulating agents in diverse practical applications, including the controlled delivery of drugs, of active substances in cosmetics, and of functional food ingredients such as enzymes [59, 96, 97]. In many cases, for example, 1 In a surfactant mixture, this threshold concentration depends on the relative composition of the surfactants.

17 )bic lo Figure 1-1: Schematic diagram of a two-component unilamellar surfactant vesicle. The two surfactant molecules are represented by the black and white heads with hydrophobic tails of different lengths. The hydrophobic region of the vesicle, which is bounded by the two dashed lines, is composed of the hydrophobic tails of the two surfactants. 17

18 that of phospholipid vesicles, the formation of vesicles requires the input of some form of energy, such as sonication [100]. These vesicles often aggregate and fuse to form large multilamellar structures within days, and are believed to be thermodynamically unstable. On the other hand, vesicles have been found to form spontaneously in some aqueous surfactant systems, including solutions containing: (i) mixtures of lecithin and lysolecithin [69], (ii) mixtures of long- and short-chain lecithins [57], (iii) mixtures of AOT and choline chloride [120], (iv) dialkyldimethylammonium hydroxide surfactants [18, 68, 129, 130, 161], (v) cationic siloxane surfactants [102], and (vi) mixtures of cationic and anionic surfactants [19, 75, 86, 87, 92]. These spontaneously-forming vesicles are believed to be thermodynamically stable in the sense that they are more resistant to aggregation and fusion, and that no energy input, besides gentle mixing, is required for their formation. 1.2 Biomedical Implication of Vesicles In addition to the industrial applications mentioned above, vesicles formed by surfactant mixtures also have a very important implication in the medical field. Besides being used as model cell membranes because of their unique closed bilayer structure [51], vesicles play an important role in the formation of cholesterol gallstones in bile Bile and Cholesterol Gallstones Human bile is formed in the liver as a solution of bile salt, phospholipid, cholesterol, electrolytes, and other components such as proteins [21]. A major fraction the bile secreted by the liver passes into the gallbladder, where it is concentrated as water is absorbed through the wall of the gallbladder. In addition to facilitating the digestion and absorption of fats, bile is also the only means by which cholesterol is excreted out of the body. The three major lipid components in bile: bile salt, phospholipid, and cholesterol, are all amphiphilic molecules, which can self-assemble to form aggregates, such as, simple micelles, mixed micelles, and vesicles, in bile [21, 111, 152]. Indeed, biliary cholesterol is solubilized by these aggregates [25], resulting in an extraordinarily

19 high cholesterol concentration in bile compared to its solubility in water [145]. Figure 1-2 depicts the general structure of a taurine-conjugated bile acid molecule. The hydroxyl groups, whose positions are indicated by R1, R2, and R3, and the ionic end of the taurine group (SO-) form the hydrophilic regions, while the fused hydrocarbon ring structure forms the hydrophobic region. The structure of a phosphatidylcholine molecule is shown in Figure 1-3. The hydrophilic head of this molecule consists of a negatively charged phosphate group, a positively charged choline group, and the glycerol backbone. It is referred to as a zwitterionic, or dipolar, head (containing both a cation and an anion). The hydrophobic group contains two hydrocarbon chains which belong to two fatty acids. The structure of a cholesterol molecule is shown in Figure 1-4. The hydroxyl group at the number 3 carbon position on ring A forms the hydrophilic moiety. Note that the cholesterol molecule has a nearly planar structure, as opposed to the buckled structure of a bile acid molecule. In lithogenic biles, cholesterol nucleates and precipitates as monohydrate crystals, which then agglomerate to form macroscopic cholesterol gallstones. Cholesterol gallstone is a common disease in most western countries. About 20 % of the population over the age of 65 have gallstones, and it is estimated that about $8 billion is spent every year on the treatment of this disease. In the past decades, significant research effort has been devoted to understanding the formation of cholesterol gallstones, and much attention has been devoted to elucidate the effects of the physico-chemical properties of bile, including the molecular structures of the lipid components (bile salt, phospholipid, and cholesterol), the composition of the bile, and the aggregation state of the lipids, on the stability of bile with respect to cholesterol nucleation [25, 94]. In particular, it has been suggested that the distribution of cholesterol between vesicles and mixed micelles, as well as the vesicle composition, play an important role in cholesterol nucleation [66, 67, 136] Model Biliary System Since native bile is such a complex system containing over forty components [21], and is not readily available, a model biliary system is often used in the experimental

20 Bile acid Cholic Acid Ursodeoxycholic Acid Chenodeoxycholic Acid Deoxycholic Acid Lithocholic Acid R1 a-oh a-oh a-oh a-oh a-oh R2 a-oh 3-OH a-oh H H R3 a-oh H H a-oh H C N/CH 2 CH 2 SO 3 I H R1 --- (b) l Hydrophobic region CHCHSO \/ CH2CH2SO 3 N H 3 R2 * Figure 1-2: Molecular structure of taurine-conjugated bile acids. (a) "Top-view" of the molecule. (b) "Side-view" of the molecule showing the buckled structure of rings A and B. The hydrophilic regions are marked by ",".

21 Hydrophilic region k Hydrophobic region A ' CH[ g :H3 C N --CH2 CH 2O- 0 P-o- H 2 2 u 0 Hy-"- O - C. S...* H C Choline group Hydrocarbon chains Figure 1-3: Molecular structure of 1-palmitoyl-2-oleoyl phosphatidylcholine. r " -- Hydrophobic region *i Figure 1-4: Molecular structure of cholesterol. The hydrophilic region is marked by ".,

22 studies of cholesterol gallstone formation. Such a system, often referred to as "model bile" in the area of bile research, consists of bile salt, phospholipid, cholesterol, water, sodium chloride, and a small amount of sodium azide to prevent bacterial growth. While native bile contains a mixture of bile salts, as shown in Figure 1-2, the "bile salt" component in a model bile is usually represented by a specific species of bile salt, such as sodium taurocholate (TC). Egg-yolk phosphatidylcholine (EYPC) is usually used to make up the "phospholipid" component because native bile contains mostly phosphatidylcholine, and the hydrocarbon chain-length distribution in EYPC is similar to that in native bile. The concentration of sodium chloride ranges from 0.1 M to 0.2 M, corresponding to the physiological concentration of sodium chloride in native bile. In addition to the components described here, other substances such as proteins or calcium salt may be included, depending on the objectives of a particular study. The phase behavior of model bile was first studied by Carey and Small [26] using various mixtures of bile salts, phospholipids, and cholesterol. They developed the equilibrium phase diagram for a ternary model biliary system having a fixed water content. A schematic representation of this phase diagram is shown in Figure 1-5. Mapping of the compositions obtained from native biles onto the phase diagram has revealed that many native bile samples fall within the three-phase region (see Figure 1-5), in which a solution of micelles and vesicles should be at equilibrium with cholesterol crystals, yet not all of them contain cholesterol crystals or gallstones [79]. This observation, together with the concept of cholesterol supersaturation index (CSI) 2, has advanced the idea of metastability in bile. The phase diagram, however, is not complete. Although the single-phase micellar solution region, that is, the region bounded by the solid line in the phase diagram (see bottom region of Figure 1-5), is quite well-defined, the boundaries involving two-phase and three-phase equilibria (regions bounded by the light dashed lines) are still unclear. 2 The cholesterol supersaturation index is a measure of the amount of cholesterol in a bile sample relative to the solubility of cholesterol in a model bile of corresponding lipid (bile salt, phospholipid, and cholesterol) composition [23].

23 1m. 20 C) '3 Phases O h Crystals / Vesicles 0, Mole Percent TC 1 nn Figure 1-5: Schematic equilibrium phase diagram for the pseudo-ternary system containing taurocholate (TC), egg-yolk phosphatidylcholine (EYPC), cholesterol (Ch), and a fixed proportion of water. The boundary of the one-phase micellar solution is denoted by the solid line for 6 g/dl total lipid content (heavy dashed line for 1 g/dl total lipid content). The two light dashed lines are the approximate boundaries separating the various multi-phase regions as indicated in the diagram. The two-phase (vesicles and micelles) metastable region is denoted approximately by the shaded region above the one-phase micellar solution. (Phase diagram adapted from Ref. 26). 23

24 1.3 Research Motivation In light of the background information given above, fundamental research on vesicle formation in surfactant mixtures will benefit many different areas, including colloid and interface science, complex fluids, encapsulation and drug delivery, and cholesterol gallstone formation in bile. The research described in this thesis addresses several important theoretical and experimental aspects of mixed vesicular systems, and the motivation for the studies to be conducted as part of this thesis is presented below Theoretical Studies of Mixed Vesicles In spite of the practical importance of mixed surfactant vesicles, as reflected in the various industrial and drug delivery applications described above, as well as in the intimate relation between vesicles and cholesterol gallstone formation in bile, there is still a lack of theoretical understanding regarding the formation of mixed surfactant vesicles. Consequently, the theoretical analysis of mixed surfactant vesicles represents an important step towards developing a better fundamental understanding of the problems encountered in the different areas cited above. More specifically, 1. In the general areas of colloid and interface science and complex fluids, vesicles represent an important class of self-assembling microstructures, as alluded to earlier. Accordingly, in order to understand the global phase behavior, as well as to rationalize the fundamental principles involved in the self-assembly of surfactant mixtures, a theoretical description of mixed vesicles is essential. In addition, as mentioned earlier, the traditional phospholipid vesicles and the spontaneously-forming cationic/anionic vesicles exhibit rather different behaviors in terms of their formation and their thermodynamic stability. This has posed challenging problems in understanding how vesicles are formed in various surfactant systems. Moreover, by carefully studying the interplay between the various free-energy contributions responsible for vesicle formation, one can also shed light on the physics and chemistry of other surfactant microstructures such as mixed micelles.

25 2. The practical implementation of vesicles as encapsulating devices in industry and in the drug delivery area also demands a more fundamental knowledge of the formation and stability of mixed surfactant vesicles. Vesicle size and size distribution, for example, play an important role in determining the amount of substances that can be encapsulated, as well as in affecting the kinetics of the release of these substances. As will be shown in chapters 4 and 5, a detailed examination of the relative importance of the various free-energy contributions associated with the process of vesiculation 3, including their interplay, can reveal valuable information regarding the factors controlling vesicle size and size distribution. 3. In the context of cholesterol gallstone formation in bile, a theoretical analysis of mixed surfactant vesicles should provide insights into the mechanism of cholesterol solubilization in bile, which may, in turn, lead to a better understanding of the problem of cholesterol nucleation in bile. Since bile can be treated as a complex fluid containing vesicles and mixed micelles, the development of a theoretical framework aimed at describing the behavior of vesicular systems can provide a starting point for the fundamental study of biliary systems. Two major theoretical approaches are currently used to study the formation of unilamellar vesicles: the curvature-elasticity approach and the molecular approach. The curvature-elasticity approach, which is by far the more popular of the two theoretical approaches, describes the vesicle bilayer as a continuous membrane characterized by the spontaneous curvature and the elastic bending modulus [72, 90]. In this approach, the formation of finite-sized vesicles depends on the interplay between these two quantities [73, 147]. The theory provides an elegant, simple way to describe the formation of vesicles, and it has been utilized in many theoretical studies to describe vesicle shape deformation and phase behavior [5, 6, 88, 119, 148, 163, 164], as well as electrostatic effects on membrane rigidity [54, 55, 117, 175]. However, because 3 "Vesiculation" refers to the process by which surfactant monomers self-assemble in an aqueous environment to form a vesicle.

26 this approach is based on a curvature expansion of the free energy of a membrane, it breaks down for small vesicles, for which the curvature is quite pronounced. In addition, within the framework of this theory, the spontaneous curvature and the elastic bending modulus are treated as phenomenological parameters, thus limiting its quantitative predictive ability. May and Ben-Shaul have recently calculated [109] these parameters for mixed bilayers using a mean-field molecular theory for chain packing, and phenomenological expressions for the head group and interfacial freeenergy contributions. They concluded that, for surfactant mixtures containing 16- and 8-carbon tails, the planar bilayer is energetically favorable, and that the addition of short tails considerably reduces the bending rigidity. Bergstr6m and Eriksson have performed similar calculations for a mixture of sodium dodecyl sulfate and dodecanol [9] using empirical expressions for chain packing and head-group interactions. They also concluded that the addition of a long-chain alcohol can significantly reduce the bending constant and therefore promote spontaneous vesicle formation. However, since these calculations are also based on the curvature-expansion approach, their conclusions are applicable only to large vesicles (small curvatures), and, therefore, the effect of surfactant tail-length asymmetry on the stabilization of small vesicles remains unclear. Nevertheless, the curvature-elasticity approach has been very successful in guiding experimental studies and explaining, at least qualitatively, many experimental observations. The molecular approach was pioneered by Israelachvili, Mitchell, and Ninham [81, 82, 115], who developed a geometric packing argument that permits one to predict the shape of self-assembling microstructures, including spheroidal, cylindrical or discoidal micelles, vesicles, and bilayers. Using a simple model based on the principle of opposing forces proposed by Tanford [162], Israelachvili and co-workers [82] predicted a near-gaussian distribution of vesicle sizes. The theory was later extended to describe two-component vesicles [30], such as those formed from mixtures of phospholipid and cholesterol, and yielded similar results. Nagarajan and Ruckenstein also developed a molecular model for vesicles [124] using a statistical-thermodynamic approach. Their model included the free-energy changes associated with the loss of

27 translational and rotational degrees of freedom of the molecules in the aggregate, and treated the electrostatic interactions between ionic or zwitterionic surfactant heads at the Debye-Hiickel approximation level [123]. Their work represents the first serious attempt to develop a predictive model for the formation of vesicles. Recently, a molecular theory based on the cell model [63] has been developed for cationic/anionic mixed vesicles [19], which predicts surface charge densities, including salt effects, in good agreement with experimental data. However, this theory does not account for the packing of the surfactant tails in the vesicle hydrophobic region, and it is not completely predictive in the sense that the vesicle radius is an input parameter determined experimentally. In order to elucidate the complex mechanism involved in the process of vesiculation, it is quite clear that a detailed molecular theory is required. A satisfactory molecular theory should be applicable over the entire vesicle size range, and allow for an estimation of the various free-energy contributions associated with vesiculation so that one can examine their relative importance and interplay in determining the vesicle properties, such as size and composition distribution. The ability to cover the entire vesicle size range is particularly important in the sense that, as mentioned in the preceding paragraph, the theoretical analysis should be capable of incorporating other microstructures, regardless of their sizes, in the study of the global phase behavior of surfactant mixtures. In developing such a theory, therefore, the various free-energy contributions associated with vesiculation must be accounted for carefully. More specifically, 1. Since the surfactant tails are constrained within the vesicle hydrophobic region, the tail packing must be treated accordingly to reflect the free-energy difference between the tails in a vesicle and those in the bulk solution. In addition, because a vesicle possesses a finite curvature, as opposed to a planar bilayer, one needs to account explicitly for the effect of curvature on the packing of the surfactant tails in a vesicle bilayer, particularly when the vesicle is very small (see chapters 2, 4, and 5).

28 2. The presence of a finite vesicle curvature poses additional challenges in the computation of the free energy of vesiculation. In particular, a curved bilayer consisting of two surfactant components requires five variables for its characterization (see chapter 2), and therefore, in the minimization of the vesicle free energy, one must sample a large configurational space. In addition, in the calculation of the electrostatic free-energy contribution, a relation between the surface potentials and the surface charge densities is required. The Poisson- Boltzmann (PB) equation can, in principle, provide such a relation, but since no analytical solution for the PB equation is available for the vesicle spherical geometry, the direct application of the PB equation, which would entail a tedious numerical integration procedure, would be quite prohibitive. Consequently, a more efficient method must be developed for the evaluation of the electrostatic free energy of a charged vesicle (see chapter 3). 3. The calculation of the steric free energy associated with the surfactant heads usually involves the estimation of the head area, which, in some cases, can be a rather ambiguous quantity. In addition, the traditional calculation of this free-energy contribution, which is based on the two-dimensional van der Waals equation of state, is known to overestimate the surface pressure at high packing densities (see discussion in chapter 2). In order to obtain a more accurate expression, therefore, an alternate formulation should be adopted in the estimation of the steric free energy (see chapter 2) Experimental Studies of Biliary System Since the nucleation of cholesterol crystals represents the initial step in a sequence of events that leads to the formation of cholesterol gallstones in bile, it will be beneficial, from a medical standpoint, to be able to identify a set of physiological variables that can most likely alter the propensity towards cholesterol nucleation in bile. As mentioned earlier, previous studies have linked the distribution of cholesterol and the vesicle composition to the metastability of bile, and therefore, it is important to

29 understand how certain physiological variables, including total lipid content, bile salt to phospholipid ratio, and cholesterol content, influence the distribution of cholesterol and the vesicle composition. In all previous studies of cholesterol distribution, however, the so-called "onevariable-at-a-time" strategy 4 [89, 135, 149, 158, 170] was used. This strategy is limited by the amount of time and materials required, and, more importantly, by its inability to identify the simultaneous effects of several variables on a particular response. In a complex system like bile, it is highly probable that physiological variables, such as those cited above, interact with each other. Accordingly, a more systematic methodology is required to provide more information, particularly with respect to the interactions between various physiological variables, on the vesicular composition, as well as on the distribution of cholesterol between vesicles and mixed micelles. A very useful and efficient way to study the simultaneous effects of a large number of variables on a particular response is through the use of statistically-designed experiments (see chapter 7). Although widely used in the chemical process industry, statistical experimental design is rarely applied in medical research. In brief, statistical experimental design is simply a systematic way of setting the experimental conditions, that is, the values of each variable under consideration. The responses at each experimental condition are measured, and a regression analysis can then be performed to estimate the coefficient associated with each variable. The values of the coefficients reflect the individual effects of the variables, as well as the interactions among them. To study the vesicle composition and the distribution of cholesterol between vesicles and mixed[ micelles, however, one needs to separate these biliary aggregates while preserving the original distribution. Two techniques are currently used to separate vesicles and mixed micelles in bile: ultracentrifugation and gel chromatography. Ultracentrifugation separates the biliary aggregates based on the difference in their densities, while gel chromatography separates them based on the difference in their sizes. Although both techniques are widely used in biliary research, there is yet no sys- 4 A "one-variable-at-a-time" strategy is one where, at each experimental condition, only one variable is changed while all the other variables are kept constant.

30 tematic comparison between these two techniques. A major problem which may have caused confusion in this area is that, in using gel chromatography, the eluant should contain the correct monomeric and simple micellar bile salt concentration, known as the inter-mixed micellar / vesicular bile salt concentration (IMC) [35, 40, 43], so that the dynamic equilibrium between the lipid monomers and the lipid aggregates can be maintained during separation. In previous studies using gel chromatography, however, an arbitrary bile salt concentration has been used in the eluant [see Ref. 40 and references cited therein], rendering the interpretation of those results very difficult. In light of these problems, a logical first step in the experimental studies of vesicle composition and the distribution of cholesterol will involve a systematic comparison between the two separation techniques, using the correct IMC in gel chromatography (see chapter 6). Depending on the outcome of this comparison, modification of the current techniques may be required in order to develop a reliable method for separating vesicles and mixed micelles (see chapter 7). 1.4 Research Objectives With the research motivation in mind, the central objectives of this thesis are twofold: 1. To develop a theoretical description of the formation of vesicles in surfactant mixtures. This objective is aimed at gaining fundamental knowledge on complex fluids in general, as well as at providing a starting point for the fundamental study of the formation of cholesterol gallstones in bile. A molecular theory for the formation of mixed surfactant vesicles will be constructed through a detailed modeling of the various free-energy contributions associated with vesiculation. This molecular theory will then be combined with a thermodynamic framework to describe the entire vesicle suspension in order to predict vesicle properties, such as, size and composition distribution, the distribution of molecules between the outer and inner vesicle leaflets, surface charge densities, and surface potentials.

31 2. To apply the statistical experimental design methodology to study the simultaneous effects of several physiological variables, such as, total lipid content, cholesterol content, and type of bile salt, on vesicle composition and the distribution of cholesterol between vesicles and mixed micelles in bile. In the development of a reliable method for separating vesicles and mixed micelles in model bile, the two current techniques, ultracentrifugation and gel chromatography, will be compared systematically to ascertain their compatibility, and, if necessary, the techniques will be modified to provide an accurate tool for the separation of the biliary aggregates. The thesis is organized as follows. In chapter 2, the details of the development of a molecular-thermodynamic theory for the formation of mixed surfactant vesicles is presented. In chapter 3, approximate expressions for the surface potentials of a charged vesicle are derived, based on the nonlinear Poisson-Boltzmann equation, which are subsequently used to evaluate the electrostatic free energy of a vesicle. In chapter 4, the theory is applied to a cationic/anionic surfactant mixture, and the theoretical predictions for this system, including vesicle size and composition distribution and surface potentials, are presented. In chapter 5, the theory is utilized to study the effect of surfactant tail-length asymmetry on the formation and stabilization of mixed surfactant vesicles. In chapter 6, ultracentrifugation and gel chromatography are compared systematically regarding their ability to separate vesicles and mixed micelles in a biliary system. In chapter 7, a modification of ultracentrifugation is developed, followed by the application of factorial experimental design to the study of cholesterol distribution and vesicular composition in model bile. Finally, conclusions and a discussion of future research directions are presented in chapter 8.

32 Chapter 2 Molecular-Thermodynamic Theory of Mixed Vesicles This chapter presents the details of the development of a molecular-thermodynamic theory to describe the formation of two-component mixed surfactant vesicles, with particular emphasis on cationic/anionic surfactant mixtures [178]. The central quantity in this theory is the free energy of vesiculation, which is calculated by carefully modeling the various free-energy contributions associated with vesiculation. By knowing only the molecular structures of the surfactants involved in vesicle formation and the solution conditions, the theory can predict a wealth of vesicle properties, including vesicle size and composition distribution, surface potentials, surface charge densities, and compositions of vesicle leaflets. A notable difference between the present theory and all the theoretical approaches mentioned in chapter 1 is in the level of detail associated with the calculation of the free-energy change of vesicle formation. More specifically: (i) the packing of the surfactant tails in the vesicle hydrophobic region is estimated through a mean-field calculation, which explicitly accounts for the conformational degrees of freedom of the tails, (ii) the electrostatic interactions between the charged surfactant heads are estimated by explicitly solving the corresponding nonlinear Poisson-Boltzmann equations, and (iii) a more accurate equation of state is adopted in the calculation of the steric repulsions between the surfactant heads. In addition, details such as the location of the outer and inner steric-repulsion surfaces

33 in a vesicle, the curvature correction to the interfacial tensions at the outer and inner hydrocarbon/water vesicle interfaces, and the existence of four charged surfaces associated with a two-component cationic/anionic vesicle are also carefully accounted for. More importantly, the theory allows for an in-depth analysis of the mechanisms of vesicle stabilization, and of the interplay between the various free-energy contributions to the free energy of vesiculation. The present molecular-thermodynamic theory also has the ability to cover the entire range of vesicle sizes (or curvatures), thus enabling a description of small, energetically stabilized, vesicles. In addition, this theory can be extended to account for the presence of other self-assembling structures possessing relatively small sizes, such as mixed micelles. The latter point is particularly important for the prediction of the global phase behavior of mixed surfactant systems that can form both mixed micelles and mixed vesicles, such as in the case of bile. 2.1 Thermodynamic Framework to Describe a Vesicle Suspension The molecular-thermodynamic theory presented in this chapter can be viewed as a generalization of the theories developed by Puvvada and Blankschtein to describe single and mixed micellar solutions [138, 139, 140]. In this theory, the total Gibbs free energy of the solution, G, is written as a sum of three contributions [10]: the standard-state free energy, Go, the free energy of mixing, Gmix, and the interaction free energy, Gint, that is, G = Go + Gmix + Gint (2.1) The chosen standard state corresponds to one in which all the surfactant monomers and the surfactant aggregates, in this case the vesicles, exist in isolation at infinite dilution, and are "fixed" in space, that is, without mixing. The free energy of mixing, Gmix, then accounts for the free-energy change due to the configurational entropy associated with mixing the aggregates, the monomers, and the water molecules. The interaction free energy, Giut, accounts for the interactions among the aggregates and

34 the monomers, which can play an important role in, for example, phase separation of a micellar solution [10, 138, 140]. In most systems in which spontaneous vesiculation has been observed, the total surfactant content is only about 1 to 2 wt% [74], and the mole fraction of vesicles in these cases can be as low as (see chapter 4). Accordingly, in the present study, it is assumed that: (i) the mixing contributing to Gmix is ideal, and (ii) the vesicle suspension is so dilute that the interaction free-energy contribution, Git, can be neglected. Of note is that the precise mathematical form of the entropy of mixing can affect the quantitative predicted size and composition distribution. In this respect, different models for the entropy of mixing have been utilized to model micellar solutions [81, 122, 124, 139], and the reader is referred to these references for further details. Consider a system containing three components: surfactant A, surfactant B, and water. Based on assumptions (i) and (ii) above, the size and composition distribution in a vesicle suspension can be expressed as follows (see appendix A for details of the derivation of Eq. (2.2)) X(n, F) = XAnF Xn(l-F) exp(-ng.es/kt) (2.2) where X(n, F) is the mole fraction of vesicles having aggregation number, n, and composition, F, which is defined as the mole fraction of component A in the vesicle, T is the absolute temperature, and k is the Boltzmann constant'. In Eq. (2.2), X1A and X1B are the mole fractions of the surfactant A and B monomers, respectively, gves is the free energy of vesiculation, defined as Yves = n,f - FPIA - (1 - F)pA/B (2.3) 1 A note of caution here is that the size and composition distribution given in Eq. (2.2) is only an approximate expression. Indeed, statistical-mechanical arguments show that, within the context of ideal mixing, a pre-exponential factor proportional to n - 1/2 should be present in Eq. (2.2) [166]. However, the value of this factor is typically very small (< 10-4 kt), compared to the uncertainties involved in the calculation of g,,,e ( 10-2 kt), and therefore can be neglected for the purpose of the present study.

35 where n, F is the standard-state chemical potential per molecule in a vesicle, and pia and 1pB are the standard-state chemical potentials of the surfactant A and B monomers, respectively. From a physical viewpoint, the free energy of vesiculation, gves, is the total free-energy change per molecule associated with the process by which nf surfactant A monomers and n(1 - F) surfactant B monomers are transferred from the aqueous environment to a vesicle having aggregation number, n, and composition, F. Equation (2.2) indicates that X(n, F) depends on the interplay of two factors: an entropic factor, XnFXn(1-F), and an energetic (Boltzmann) factor, exp(-ngves/kt). The entropic Jfactor reflects the penalty associated with localizing the surfactant A and B monomers at a certain position in space, that is, in a vesicle, while the energetic factor reflects the propensity of the surfactant A and B monomers to aggregate. 2.2 Molecular Model of Vesicle Formation To evaluate the vesicle size and composition distribution, X(n, F) in Eq. (2.2), one needs an explicit model for the free energy of vesiculation, gv,,e. The free energy of vesiculation can be viewed as composed of the following five contributions: (1) the transfer free energy, gtr, (2) the packing free energy, g9pack, (3) the interfacial free energy, go, (4) the steric free energy, gsteric, and (5) the electrostatic free energy, g,~ec (see Figure 2-1). Mathematically, one can therefore write gves = gtr + gpack + ga + gsteric + gelec (2.4) These five free-energy contributions account for the essential features that differentiate a surfactant molecule in the vesicle and in the monomeric state. The transfer free energy, gtr, reflects the so-called hydrophobic effect [162], which constitutes the major driving force for surfactant self-assembly in water. Indeed, the transfer free energy is the only favorable free-energy contribution to molecular aggregation, with the other four free-energy contributions described in Eq. (2.4) working against this process. The hydrophobic region in a vesicle, however, is different from bulk hydro-

36 J elec steric / pack S gtr LII '0 W I I I I I I I I Aque Regih IL tb IL1 b IL Hydrc \Regic -- ft /I, Aqueous Region / I Figure 2-1: Schematic representation of a two-component (represented by the black and white heads) unilamellar vesicle showing: (i) part of the hydrophobic bilayer region composed of the surfactant tails of both surfactant species, (ii) the location of the outer and inner hydrocarbon/water interfaces, and (iii) the various regions with which the free-energy contributions, gtr, gpack, ga, gsteric, and gedec, are associated.

37 carbon. In a vesicle, the surfactant tails are anchored at one end on either the outer or inner interfaces, which restricts the number of conformations that each surfactant tail can adopt while still maintaining a uniform liquid hydrocarbon density in the vesicle hydrophobic region. This subtle difference between a bulk hydrocarbon phase and the hydrophobic region in a vesicle is captured by the packing free energy, gpack. In addition, free-energy penalties are imposed, upon aggregation, by the creation of the outer and inner hydrocarbon/water interfaces, captured in g,, and by the steric repulsions and electrostatic interactions between the surfactant heads, captured in gsteric and gelec, respectively. The following paragraphs briefly describe each freeenergy contribution, including their estimation based on knowledge of the molecular structures of the surfactant molecules involved in vesicle formation and the solution conditions Transfer Free Energy The process by which the surfactant tails are transferred from the aqueous environment to the hydrophobic vesicle bilayer upon aggregation can be viewed as being composed of three steps: (i) the surfactant tails are transferred from the aqueous environment to their corresponding pure hydrocarbon phases, (ii) the surfactant tails are then mixed to form the outer and inner hydrocarbon mixtures, corresponding to the outer and inner "leaflets", or monolayers, that constitute the vesicle bilayer, and (iii) the surfactant tails are anchored at one end on the outer or inner vesicle interfaces (see Figure 2-1). The free-energy change associated with the third step can be accounted for by the packing free energy, gpack, as will be discussed in (2) below. The free-energy changes associated with the first two steps are captured by the transfer free energy, gtr. Accordingly, for a vesicle having aggregation number, n, and composition, F, gtr can be expressed as gtr = FA/1tr,A + (1 - F)Al-tr,B + 9m (2.5)

38 where AnPtr,A and APtr,B are the free-energy changes associated with transferring the tails of surfactants A and B from the aqueous environment to their corresponding pure liquid hydrocarbon phases, respectively, and gm is the free-energy change per molecule due to mixing of the tails of surfactants A and B in the outer and inner vesicle leaflets. Strictly speaking, because of the proximity of the surfactant heads in a vesicle, the environment surrounding a surfactant head in a vesicle can also be different from that in the aqueous environment. However, the effect on vesiculation due to this difference is likely to be much smaller than that caused by transferring the surfactant tails, particularly for long-chain hydrocarbons, and it is therefore reasonable to neglect this difference as a first approximation. The free-energy change, APtr,k (k = A or B), arises mainly from the rearrangement of water molecules surrounding the surfactant tails when they are transferred from the aqueous environment to the pure hydrocarbon phase. This free-energy change can be estimated directly from solubility data, since the process of dissolution can be viewed as the reverse of the transferring process described above. In particular, for alkyl tails, empirical relations based on experimental solubility data are available, which express APtr,k as a function of carbon number and temperature [1]. Specifically, A/tr,k,,= ( n,k) ( nc,k) (2.6) kt T where n,,k is the carbon number of the tail of component k (A or B) in the hydrophobic region. In this formulation, the hydrocarbon/water interface is located between the first and second carbon atoms in the tail. Accordingly, n,,k should be one less than the total number of carbon atoms in the tail. For example, for a surfactant tail containing 16 carbon atoms, such as that corresponding to cetyltrimethylammonium bromide (CTAB), n,,k is equal to 15. This choice of the location of the interface is mainly due to possible water penetration into the hydrophobic region [138]. In other words, the first carbon atom of the tail is allowed to come into contact with water. The free-energy change per molecule associated with mixing the tails of surfactants A and B in each vesicle leaflet is estimated using ideal mixing as a first approximation,

39 that is T= f Z Xok lnxok + (1- f) Z X1iklnXik (2.7) k=a,b k=a,b where Xok and Xik are the mole fractions of component k (k = A or B) in the outer and inner leaflets, respectively, and f is the mole fraction of surfactant molecules in the outer leaflet, that is, Number of surfactant molecules in the outer leaflet = Total number of surfactant molecules in the vesicle (2.8) The mole fractlion, f, thus characterizes the distribution of surfactant molecules between the outer and inner leaflets in a vesicle. As will be shown in chapter 4, f is perhaps the most important variable affecting the thermodynamics of vesiculation Packing Free Energy In the hydrophobic region of a vesicle, the surfactant tails are anchored at one end on the outer and inner vesicle interfaces, which impose restrictions on the conformations of the surfactant tails. This packing penalty is captured in this molecular model by the packing free energy, gpack, which is estimated as the free-energy difference between a surfactant tail packed in a vesicle and a surfactant tail dispersed in bulk hydrocarbon, that is, f (2.9) gpack = Ipack - pack (2.9) where Ipack is the free energy per molecule due to packing of a surfactant tail in a vesicle, and fpack is the packing free energy corresponding to a "free" surfactant tail (see appendix B). In this study, the mean-field approach developed by Szleifer and co-workers is generalized [160] to calculate this free-energy contribution. Briefly, ppack can be written as follows Ipack = Z [fxoklok + (1 - f)x i k ik] (2.10) k=a,b

40 where Pok (Iik) is the packing free energy per molecule of component k in the outer (inner) leaflet, which can be written in terms of the single-chain probability distribution of chain conformations. For example, for component k in the outer leaflet, one has Pok = P(ak)Ek(cak) - ktu : P(ak) in f(ak) (2.11) where P(ck) is the probability of component k in the outer leaflet adopting a conformation, ak, and Ek(&k) is the internal energy of the chain corresponding to the conformation, ck. The probability, P(ck), can be related to the volume of the hydrophobic region through the following relation where (ok E [f X o k(ok(r)) + (1 - f)xik(oik(r))1 = a(r) (2.12) k=a,b (r)) and ( ik(r)) are the configurational-average segment volume densities (volume per unit length) at position, r, due to component k in the outer and inner leaflets, respectively. For example, for component k in the outer leaflet, one has (ok(t)) = P(ck) ok (ak, r) (2.13) ak A similar expression can be written for ( ik(r)). The quantity, a(r), in Eq. (2.12) is the volume density available at position, r. Here, the density of the vesicle hydrophobic region is assumed to be uniform and equal to that of liquid hydrocarbon. Accordingly, for given values of f, Xok, Xik, Ro, and Ri, a(r) depends only on the geometry of the vesicle, and the probability distribution can be obtained by solving Eq. (2.12). The constraint of uniform liquid density in the vesicle hydrophobic region should be a valid assumption based on a comparison with experimental observations [61, 62], although simulations of phospholipid bilayers have shown that the density may decrease towards the center of the bilayer [107]. This constraint can, in fact, be relaxed, provided that the density profile in the hydrophobic region is known. However, using an explicit non-uniform density profile in this calculation will certainly introduce some ambiguities, since the profile is not known a priori in most cases.

41 Consequently, rather than using the density profile as an arbitrary parameter, the density is kept uniform in all the calculations which follow. Knowing P(ak), Ppack can then be calculated using Eqs. (2.10) and (2.11). The general procedure for solving Eq. (2.12), including the discretization of the vesicle hydrophobic region, can be found in Refs. 159 and 160, and it will not be detailed here. However, some useful formulas that are specific for the vesicle geometry are presented in appendix B. These formulas should be helpful to readers who are interested in actually performing such calculations. A noteworthy point here is that, instead of treating the vesicle bilayer as a planar bilayer, the theory explicitly accounts for its curvature. Consequently, five variables (f, Xok, Xik, Ro, and Ri) are needed to characterize a vesicle bilayer in the packing calculations, as opposed to only two variables (thickness and composition) in the planar case 2. In the present study, the packing free energies are generated for a fixed number of combinations of these five variables, and for other combinations of these variables, the corresponding packing free energies are obtained by interpolation. This somewhat tedious packing free-energy calculation is necessary to ensure the applicability of this theory to the entire range of vesicle sizes. As will be shown in chapter 4, the effects of curvature and the freedom with respect to the distribution of surfactant molecules between the two vesicle leaflets, reflected in the variable, f, are particularly important as the vesicle size becomes small. In a rigorous calculation of the free energy of vesiculation, then, the packing contribution must be able to reflect these effects in the small vesicle size range. Nevertheless, for systems that are dominated by large vesicles, the packing free energy may be approximated by that corresponding to a planar bilayer without a significant loss of accuracy Interfacial Free Energy As the surfactant molecules self-assemble to form the vesicle, the outer and inner interfaces between the hydrophobic region and the aqueous environments are created 2In a planar bilayer, f = 0.5, Xok = Xik, and the absolute values of Ro and Ri are irrelevant, since only their difference, that is, the planar bilayer thickness, is important.

42 (see Figure 2-1). The free-energy change per molecule, g 0, required to create these two interfaces can be captured in the following expression g, = f o(ao - ao) + (1 - f))&(ai - ai) (2.14) where ao (a 2 ) and o (a*) are the area per molecule and the molar-average shielded area per molecule at the outer (inner) interface, respectively. The shielded area is the area occupied by the surfactant tail at the interface, and it reduces the area of contact between the hydrophobic region and water. The molar-average shielded areas are defined as follows S= Xoka* (2.15) k=a,b i= Xika* (2.16) k=a,b where a* is the shielded area of component k. In Eq. (2.14), ao and ai are the curvature-corrected molar-average interfacial tensions at the outer and inner interfaces, respectively. The curvature correction to the interfacial tension can be estimated using the Tolman equation [167]. In particular, for the range of vesicle sizes considered here, o = op- 2) (2.17) Ro &j = i( - 2) (2.18) Ri where 6 is the Tolman length, and Ro (Ri) is the outer (inner) vesicle radius, measured from the center of the vesicle to the outer (inner) hydrocarbon/water interface. In Eq. (2.17) (Eq. (2.18)), op (ip) is the molar-average planar interfacial tension associated with the outer (inner) interface, estimated, as a first approximation, as op = Z Xokuk (2.19) k=a,b

43 i = Xik Ok (2.20) k=a,b where Uk is the planar interfacial tension between component k (k = A or B) and water. If, for,example, component A consists of a 16-carbon tail and component B consists of a 8--carbon tail, then cra and abs should be the interfacial tensions between water and pentadecane, and water and heptane, respectively 3. The effect of curvature on interfacial tension is, in fact, not a trivial issue. Indeed, significant research effort has been devoted since the derivation of Eqs. (2.17) or (2.18) by Tolman [167], particularly regarding the estimation of the Tolman length,6[52, 77, 91, 126, 144, 1.67]. The simplest definition of the Tolman length, 6, is the distance between the surface of tension and the Gibbs dividing surface. The existence of a finite curvature results in a non-zero value of this distance, as opposed to zero in the planar case, and effectively enhances or reduces the interfacial tension from its planar value, depending on the sign of 6, or more specifically, depending on the densities of the two phases involved. Because the Tolman length, 6, only comes into play in systems containing droplets having very small sizes, it is usually difficult to obtain an accurate experimental measurement of 6. Theoretical studies and simulations of Lennard-Jones fluids have set the value of 6 between -0.2d to -0.4d for droplets, where d is the hard-sphere diameter [52, 70, 83, 127]. In this study, 6 is estimated to be 1.4 A and -1.4 A for the outer and inner interfaces, respectively. The estimation is based on a linear density profile across an interfacial region having a thickness of about 2.5 A, which corresponds approximately to twice the projected length of a carboncarbon bond, in accordance with the water-penetration region (see section 2.2.1). The linear profile is just a simplifying assumption; indeed, other profiles, such as sigmoidal, have yielded no significant difference. Interestingly, the estimated 6 value of 1.4 A agrees well with the simulation results discussed earlier, if we treat the hydrophobic region as composed of "spheres" of methylene segments (d e 4 A). It is beyond the scope of this work to provide a thorough investigation on the Tolman 3 Recall that, as stressed earlier, the number of carbons in the hydrophobic region is one less than that in the actual surfactant tail.

44 length. Instead, we treat 6 as a fixed parameter which reflects the influence of finite curvature in the calculation of the interfacial free energy, g,. An interesting point to note here is that the Tolman length for the inner interface has a sign opposite to that for the outer interface because the phases involved (hydrocarbon and water) are reversed in this case. Consequently, the effect of curvature works to reduce the interfacial tension at the outer interface, whereas it increases the interfacial tension at the inner interface Steric Free Energy Surfactant heads have a finite size, and therefore, when they are brought together to form a vesicle, the steric repulsions between these heads will invariably incur a freeenergy penalty to the process. This steric free-energy contribution can be estimated as the free-energy change associated with the process by which the surfactant heads are brought from infinitely apart to the state corresponding to the vesicle interfaces. For example, for the outer interface, this can be expressed as Gsteric,o -= (Hn- jid)dao (2.21) where Gsteric,o is the total outer steric free energy, Ao is the total area of the outer steric-repulsion surface, H is the surface pressure, and H id is the ideal surface pressure. The outer (inner) steric-repulsion surface is defined as the surface located at a distance, dch,o (dch,i), from the outer (inner) hydrocarbon/water interface, where dch,o = Xokdch,k (2.22) k=a,b dch,i = Xikdch,k (2.23) k=a,b and dch,k is the charge distance of component k, which is the distance between the location of the charge in the head of component k and the hydrocarbon/water interface. An expression similar to Eq. (2.21) can be written for the total inner steric free energy, Gsteric,i, and the total steric free energy of the vesicle is simply

45 Gsteric = Gsteric,o + Gsteric,i. In the present theory, it is assumed that the surfactant heads are compact, so that they can be modeled as hard disks which are characterized by fixed diameters. Note, however, that for chain-like surfactant heads such as those of the poly(ethylene oxide) variety, this hard-disk approach is probably not applicable since the heads may be quite flexible in that case. The treatment of flexible surfactant heads is beyond the scope of the present study, and the interested reader is referred to Ref. 22 and references therein for a description of various ways to deal with such flexible heads. Traditionally, the two-dimensional repulsive van der Waals (vdw) equation of state has been used to evaluate Gteric [19, 121, 123, 124, 125, 139, 140]. Using this equation of state to express the surface pressure, II, in Eq. (2.21), one would obtain the familiar logarithmic form of the steric free energy. The problem with using the vdw equation of state is that the estimation of the so-called head area of the surfactant molecule is quite ambiguous. Theoretically speaking, in order to be consistent with the description of the vdw equation of state, the head area should really be the excluded area per molecule. However, estimates of this quantity vary within a wide range, even for a simple surfactant head such as a sulfate [19, 121, 123, 124, 125, 139, 140]. In addition, the vdw equation of state is known to overestimate the surface pressure for a hard-disk system [112], particularly when the hard disks approach a high packing density. This behavior may prevent the surfactant heads from coming too close to each other in a vesicle, thus resulting in an overestimation of the area per molecule. To overcome this difficulty, and in an attempt to obtain a more accurate expression for Gsteric, the scaled-particle theory (SPT) equation of state for hard-disk mixtures [56, 98, 128] is used in the present study. In addition to its simplicity, the choice of the SPT equation of state is mainly due to two reasons: (i) in the SPT equation of state, it is the hard-disk area that comes into the formulation, thus eliminating the ambiguity discussed above, and (ii) the behavior of surface pressure at high packing densities is more realistic than that predicted by the vdw equation of state, which should result in a more reliable estimate of the area per molecule. Performing the integration in Eq. (2.21) using the SPT equation of state

46 (see appendix C for details), one obtains 9sterc =fin _ /4 1 -n ho + - f(1 ) I/4 - In 1 - i (2.24) kt fa' - aho a/ a - ah i a/ where gsteric = Gsteric/n is the steric free energy per molecule, do (di) and aho (ahi) are the molar-average hard-disk diameter and hard-disk area (see appendix C for definitions) of the surfactant heads at the outer (inner) interface, respectively, and a' o (a') is the outer (inner) area per molecule calculated at the outer (inner) steric-repulsion surface. Note that a' and a' are different from ao and ai used in Eq. (2.14), which are the area per molecule at the outer and inner hydrocarbon/water interfaces, respectively. Indeed, a' and a' are related to ao and ai through geometric considerations by the following relations a' o = ao 1+ ho (2.25) - 2 a = ai 1 - (2.26) This correction to the area per molecule reflects the fact that the steric repulsions between the surfactant heads occur at slightly different locations away from the outer and inner hydrocarbon/water interfaces. As in the case of the curvature correction to the interfacial tension (see Eqs. (2.17) and (2.18)), this correction becomes significant only in the case of small vesicles Electrostatic Free Energy As stated earlier, vesicles form spontaneously in certain mixtures of cationic and anionic surfactants. In order to account for the electrostatic interactions between the oppositely-charged surfactant heads in the vesicle, one needs to calculate the electrostatic free energy, gelec, which acts to oppose the self-assembling process. Several methods may be used to estimate geiec, including the simple "capacitor" model [30, 81, 82], and the calculation of the internal energy and the entropy of demixing the

47 ions in aqueous solution [63, 106]. A more direct approach involves the calculation of the reversible work required to charge all the surfaces involved. In a two-component vesicle, there can be four such surfaces, since the distances, dch,a and dch,b, need not be the same. Accordingly, a rigorous calculation requires charging the four surfaces simultaneously (see Figure 2-2(a)). To charge a surface, the relation between the surface potential and the surface charge density must be known. The Poisson-Boltzmann (PB) equation provides such a relation, but there is as yet no analytical solution to the PB equation in spherical geometry. Consequently, the direct application of the PB equation to the charging process can be tedious since a numerical solution is required at each charging step. Furthermore, as will be discussed in section 2.3, the configuration of an isolated vesicle, which is characterized by such variables as the distribution of molecules, f, the outer and inner leaflet compositions, Xok and Xik, and the thickness of the hydrophobic region, tb, is obtained by minimizing the free energy of vesiculation with respect to these variables. Such a minimization procedure will invariably sample a large configuration space, which makes the numerical solution of the PB equation quite prohibitive. To simplify the calculation of gelec, an approximate charging approach is adopted in the present study. Instead of charging the four surfaces simultaneously as shown in Figure 2-2(a), we estimated gelec as the free energy corresponding to charging an outer and inner spherical capacitor, depicted in Figure 2-2(b), plus that corresponding to placing the net charges on two surfaces, depicted in Figure 2-2(c). The outer spherical capacitor consists of the two surfaces defined by R 3 and R 4, with an electric charge of Q3f, while the inner spherical capacitor is composed of the surfaces defined by R 2 and R 1, with an electric charge of Q2f. Mathematically, therefore, g9elec can be expressed as follows ngelec Q2f D +=Q Q2 3fD D + o(aqda (2.27) 2, R +(1 + D/R1) 2cER2(1 + D/R 3 ) where Q'o = Q3f + Q4f (Ql = Qlf + Q2f) is the final net charge on the outer (inner)

48 0...I~ -4T / I I I 3 f I \\ (a) Four-Surface Configuration o = Q 3 f + Q4f (b) Capacitor (c) Net Charges Figure 2-2: Schematic diagram depicting the approximation used in the calculation of the electrostatic free energy, geec. The four-surface configuration in (a) is replaced by a configuration that consists of an outer and inner capacitor in (b), plus the net charges on the outer and inner interfaces in (c). The charge on each surface is denoted by Qj,J -= 1,..., 4, and D is defined as D = R 2 - R1 R 4 - R 3. 48

49 charged surface 4, Vo (Vi) is the outer (inner) surface potential, Qjf, j = 1,..., 4 is the final charge at Rj, c, is the permittivity in water, A is the charging parameter, and D is the so-called gap distance, which is defined as D = Idch,A - dch,bi (2.28) The final charge on each surface can, of course, be calculated by knowing the aggregation number, n, the distribution of molecules, f, and the mole fraction of each component. For example, Q3f can be expressed as Q3f = nfxoaez, where e is the elementary charge and z is the valence of component A 5. The first two terms on the right-hand side of Eq. (2.27) are the inner and outer capacitor terms, and the integral term corresponds to the charging of the inner and outer charged surfaces from zero to the total net charges. The derivation of Eq. (2.27), including the approximations involved in this approach, can be found in appendix D. Note that the charged surfaces are not the same as the hydrocarbon/water interfaces, which are located at Ro and Ri (see Figure 2-1 and section 2.2.1), nor are they the same as the steric-repulsion surfaces, which are located at dch,o and dch,i (see section 2.2.4). To further facilitate the calculation of gelec, approximate expressions have been developed for the two surface potentials, Vo and ji, based on the nonlinear PB equation. The derivation of these expressions is given in chapter Computational Procedure Many equations are involved in the present molecular-thermodynamic model for the description of mixed cationic/anionic vesicles. Before we proceed to discuss the com- 4 Note that the term "final", as used here, does not imply the minimum-energy configuration of the vesicle. Instead, it refers to the charging stage in the calculation of gelec. In other words, the term "final" corresponds to the state at which A = 1. As discussed in section 2.3, ges is minimized by sampling a large configuration space, and each sampled vesicle configuration will have a "final" charge on each surface. 5 Here, it is assumed that the distance between the location of the charge on the surfactant head of component A and the hydrocarbon/water interface, dch,a, is smaller than that corresponding to component B, dch,b.

50 putational procedure involved in the implementation of these equations, it will be beneficial to the reader to present a summary of these equations. Such a summary is given in appendix E, in which references to other expressions required in these model equations are also provided. A total of variables are involved in the calculation of g,,,. In addition to n and F, there are two areas per molecule, ao and ai, the distribution of molecules between the two leaflets, f, the composition of each leaflet, XoA and XiA, the outer and inner radii, Ro and Ri, and the thickness of the hydrophobic region, tb (see Figure 2-1). These variables are not totally independent, but are, instead, related through constraints imposed by the geometry of the vesicle. Indeed, as shown in appendix F, there are five such geometric relations among the ten variables cited above. These geometric relations can be used to eliminate five variables. Specifically, at each n and F, one can calculate the free energy of vesiculation by minimizing g,,,es, as given in Eq. (2.4), with respect to three independent variables: XoA, f, and tb. The choice of these three variables is mainly based on the convenience in solving the geometric constraints. From a physical standpoint, this procedure of calculating the free-energy surface implies that, at given values of n and F, an isolated vesicle will seek a minimum freeenergy configuration, and it will not be affected by the presence of other vesicles in the suspension. This procedure is valid under the assumption of negligible inter-vesicular interactions, captured in Gint, as stated in section 2.1. If Gint is not negligible, however, the minimum free-energy configuration of an isolated vesicle may be different from that of a vesicle in suspension, since the interactions may depend on vesicle size, which, in turn, may be influenced by the vesicle configuration. In this case, then, the free energy of the entire vesicular suspension, G, should be minimized (see Eq. (2.1)). Since one can now compute gves at every n and F, Eq. (2.2) can be used to express X(n, F) as a function of X1A and X1B. This relation can then be inserted in the mass balance equations, which state that, for a two-component vesicular system, XAt = X1A + n FX(n,F)dF (2.29) n

51 XBt = X1B + n (1 - F)X(n, F)dF (2.30) n where XAt and XBt are the total mole fractions of components A and B in the system, respectively. Since XAt and XBt are experimental inputs, the only unknowns in Eqs. (2.29) and (2.30) are the monomer mole fractions, X1A and X1B, which can be solved for by using a simple trial and error procedure. After obtaining the monomer mole fractions, X1A and X1B, the quantity X(n, F) can be calculated directly using Eq. (2.2). A noteworthy point here is that, in the calculation of gel,,, the ion concentration, which plays an important role in the screening of the surface potentials, includes the monomer concentrations, X1A and X1B. This implicit relation calls for an iterative procedure, thus making any rigorous calculation rather tedious. In some cases, however, one may be able to make certain approximations so as to simplify this calculation. For example, when there is a large amount of added salt present in the system, compared to the surfactant concentration, the concentration of added salt will simply swamp out the monomer concentration, rendering it insignificant as far as the calculation of gelec is concerned. 2.4 Concluding Remarks In summary, this chapter has presented a thermodynamic framework to describe a vesicle suspension, and discussed thoroughly the estimation of the various free-energy contributions to the free energy of vesiculation, g,es. When compared to previous molecular approaches, this theory provides a more precise account of the various freeenergy contributions to vesiculation, including an evaluation of the packing free energy associated with the surfactant tails and the steric free energy associated with the surfactant heads. Unlike the widely used continuum or curvature-elasticity approach, the present theory accounts explicitly for the molecular nature of the aggregates, and therefore provides many more details about the configuration of the vesicles over the entire range of radii or curvatures. In chapter 4, this theory will be applied to study vesicle formation in a cationic/anionic surfactant mixture. In addition to

52 demonstrating the ability of the theory to predict vesicle properties, such as, size and composition distribution and surface potentials, the study presented in chapter 4 also illustrates how the interplay between the various free-energy contributions associated with vesiculation affects the formation of mixed vesicles. The theory will then be utilized in chapter 5 to study the effect of surfactant tail-length asymmetry on the formation and stabilization of mixed surfactant vesicles. As stated in section 2.2.5, however, in order to estimate the electrostatic free energy, gelec, of a charged vesicle via the reversible charging process, a relation between the surface potentials and the surface charge densities must be established. Therefore, before proceeding with the applications of the theory, the following chapter will first detail the derivation of the approximate analytical expressions used in the calculation of the surface potentials of a charged vesicle.

53 Chapter 3 Approximate Expressions for the Surface Potentials of Charged Vesicles In this chapter, approximate relations between the surface potentials and the surface charge densities are derived for the purpose of evaluating gele of a charged vesicle [177]. The surface potentials of a charged vesicle may, in principle, be calculated by solving the Poisson-Boltzmann (PB) equation (see section 3.1). Unfortunately, an analytical solution of the PB equation in spherical geometry is not yet available, and, therefore, an often tedious numerical integration procedure is required [50, 113]. Consequently, it is quite prohibitive, from a computational standpoint, to utilize the PB equation in the minimization of g,,,es. In this respect, several approximate analytical solutions of the PB equation have been developed for a single charged sphere in an electrolyte solution [8, 108, 132, 174]. In particular, Evans, Mitchell, and Ninham (EMN) derived [48, 49, 116] an analytical expression for the electrostatic free energy of ionic micelles in their development of the dressed-ionic micelle theory. Mitchell and Ninham later extended [117] this formulation to charged vesicles, where they assumed that the interior of the vesicle is electrically neutral, and that the electrostatic potential at the center of the vesicle is zero. Although these assumptions simplify the mathematical complexities, they can be restrictive under conditions of

54 low ionic strength and/or small vesicle radius where the potential in the interior of the vesicle may not decay to zero at the center. In addition, the assumption of an electrically neutral interior implies that the electrostatic potential does not vary across the hydrophobic region, which is valid only when the vesicle has similar outer and inner surface charge densities. In the present derivation of approximate expressions for the surface potentials, no assumption of zero center-point potential and electroneutrality in the interior of the vesicle is made. Consequently, a solution strategy different from that of EMN is required, since the outer and inner surface potentials are coupled through the potential profile in the hydrophobic region. The derivation of the approximate relations is presented in two stages. First, in section 3.1, we derive a set of three approximate algebraic equations describing the relations between the surface potentials, the center-point potential, and the surface charge densities, based on a generalization of the approach of EMN. The two boundary conditions at the outer and inner surfaces of the vesicle serve as the backbone of this derivation. This set of equations can then be solved numerically, and the resulting surface potentials can be used in Eq. (2.27) in chapter 2 to evaluate the electrostatic free energy of the charged vesicle. In the second stage (see section 3.2), other approximations are introduced in order to obtain analytical expressions for the surface potentials. Using these analytical expressions, the surface potentials can be calculated directly without any numerical procedure, and, therefore, gelec in Eq. (2.27) can be calculated much more efficiently. Accordingly, the derivation in the second stage represents an additional improvement on the efficiency of utilizing Eq. (2.27), as compared to simply utilizing the approach of EMN. A detailed derivation of the analytical expressions for the vesicle surface potentials is presented in the appendix G.

55 3.1 Implicit Relations between Surface Potentials and Surface Charge Densities In what follows, the charged vesicle is modeled as composed of three regions, which are separated by two charged surfaces (see Figure 3-1). Regions 1 and 3 are the aqueous domains containing water and ions, and Region 2 is the hydrophobic domain made up of the surfactant tails. It is assumed that the ions can cross freely, but not accumulate in, the hydrophobic region [50, 113, 165]. Assuming that both the surfactant and the added salt are symmetric electrolytes having the same valence, z, and that the system is spherically symmetric, the nonlinear PB equation for each of the three regions can be written as follows 1. Region 1 (0 < r < Ri): d2y1 d 2 Y 2 dyl 1 + = sinh(yi) (3.1) dz 2 x dx 2. Region 2 (R, < r < Ro): 3. Region 3 (r > Ro): d 2 y dy + 2 d = 0 (3.2) dz 2 x dx d 2 y 3 2 dy dx d sinh(y 3 ) (3.3) dx2 x dx where yj = ez/kt, j= 1, 2, 3 (3.4) x = Kwr (3.5) 87rTno e 2 z 2 = 8W6W (3.6) In Eqs. (3.1) - (3.6), Oj is the electrostatic potential in Region j (1, 2, or 3), yj is the reduced potential in Region j, r is the radial coordinate, w, is the inverse of the Debye screening length, ce = 47r1wo is the permittivity of water, where ry, is the dielectric constant of water and eo is the permittivity in vacuum, k is the Boltzmann

56 Region 3 (Outer Aqueous) + -F K + Figure 3-1: Schematic diagram of a positively-charged vesicle showing the inner and outer aqueous regions, separated by the hydrophobic region composed of the surfactant tails. The vesicle is assumed to be spherical, and the charges are assumed to be smeared on the surfaces at Ri and Ro, the inner and outer radii, respectively.